State representation of position in 2D plane for Reinforcement Learning (Q Learning)
I think your net should have the various actions as outputs, but I am not an expert in Deep Nets. I just think that that light form of multi-task learning might be better. The idea of multi-task learning is that a predictor predicting multiple variables (in this case the various Q(s,a1), Q(s,a2), ...) using mostly the same structure (varying only the output weights) will learn more sensible things. Though I admit applying this here might be a bit of a stretch.

As for the real question, a popular technique in Reinforcement Learning is Tile Coding.

The basic idea is to discretize the (2-dimensional, in your case) state-space - imagine a grid laid over the 2D space - and assigning an input feature to each cell; all of these variables are set to zero except for the one your continuous variables fall into. For example, if your grid is 20x20, you will have 400 variables, 399 of which are set to zero, and 1 set to one.

Tile Coding takes this one step further and repeats this using slight offsets for the grid. Imagine you create an identical grid but you move it slightly to the right by 1/10 of the width of a cell: you will have another set of 400 variables like before, but it is possible that the cell set to one is not the same. Then you repeat this moving the grid by 2/10 and you have another set of 400 variables, again, only 1 of which is set to one. In total you have 10 sets of 400 variables (if you repeat more than that, you get the same grids as before); of your 4000 variables, only 10 are set to one.
Now you repeat this by adding a 1/10 of a cell offset in the Y axis and obtain another 4000 variables. Repeat with 2/10 and you get another 4000. By the end of it, you have 40000 variables, 100 of which are set to one.

Now your net can more easily learn different weights for different positions. I recommend you to follow the link above for a better explanation than mine (and figures!)

My suggestion is to feed all of these variables to your net and have it predict the Q-value for all of the actions. But, again, I am no expert in deep nets so I may be wrong.

Also, according to Andrej Karpathy, "most people prefer to use Policy Gradients, including the authors of the original DQN paper who have shown Policy Gradients to work better than Q Learning when tuned well.". This means that you may be better off not using Q-learning (as they did in the original DQN formulation) to train your net. Have a look at Andrej's blog and the paper he points to.